Learn Romanian: Vara devine din ce în ce mai caldă. - Summer is getting hotter and hotter.

Yes! Vara devine tot mai caldă means exactly the same thing. Tot mai is just a shorter, often slightly more conversational way to express this "more and more" progressive comparison than din ce în ce mai.

Why is the adjective caldă used instead of the masculine form cald?

The adjective must agree with the noun it describes. Since vara (the summer) is a feminine singular noun, the adjective takes the feminine singular form caldă. Even though the adjective is separated from the noun by the verb and the comparison phrase, this agreement rule still applies!

In English we say "is getting", but here the Romanian verb is devine. Why?

In English, "to get" has many meanings, including "to become". In Romanian, you cannot translate this change of state with verbs like a primi or a obține (which mean "to get" in the sense of receiving). Instead, you must use a deveni, which specifically means "to become". Literally, the sentence says "The summer becomes more and more hot."

Why does the sentence start with Vara (the summer) instead of just Vară (summer)?

In Romanian, when you make a general statement about a concept, season, or category and it acts as the subject of the sentence, you almost always need to attach the definite article. Therefore, vară (summer) becomes vara (the summer).